Beat egg yolks and sugar in the bowl of electric mixer fitted with whisk attachment until thick and pale and they form thick ribbons (5 to 8 minutes); add mascarpone; beat until smooth; mix in 1 Tbsp. espresso; place 1/2 of mixture into another bowl; fold in persimmon puree; cover both with plastic wrap; reserve both in refrigerator.

Mix remaining espresso and rum in a small shallow dish; quickly dip each ladyfinger in espresso mixture (do not soak for too long or they will fall apart); place in a single layer along bottom of large serving bowl or baking dish to cover completely (break cookies if necessary).

Spoon 1/2 persimmon mixture over ladyfingers; smooth with offset spatula; top persimmon mixture with 1/2 mascarpone mixture; smooth; repeat layers one more time with remaining ladyfingers, persimmon mixture, and mas­car­pone mixture; cover with plastic wrap; reserve in refrigerator for at least 2 hours or overnight.